J. B. Solie*, W. R. Raun, and M. L. Stone
Dep. of Biosystems and Agricultural Engineering and Dep. of Plant and Soil Sciences, Oklahoma State University, Stillwater, OK 74078 and The Samuel Roberts Noble Foundation, Ardmore, OK; Contribution from the Okla. Agric. Exp. Stn. *Corresponding author
The resolution at which prescribed applications should be made in agricultural production is not well defined. This study was conducted to determine the semivariance range (zone of influence of the sample) where soil test and plant uptake measurements were related from previously published analytical data. Soil and plant analyses were performed in 490-0.3 x 0.3 m plots from bermudagrass pastures at two locations. Eight soil cores (0 to 15 cm deep) were collected and composited from each 0.3 x 0.3 m plot. Semivariance analysis was used to estimate the range over which samples of the five soil variables (total N, extractable P, and K, organic C, and pH) and two plant variables (N concentration and biomass) were related. Semivariance statistics including: the nugget, sill, N:S ratio, range of relatedness were calculated. A linear-plateau (linear to a sill) function or a peak function was fitted to the data to determine the range of relatedness between variable measurements. The range from semivariance analysis was generally between 1.04 and 6.70 m, but was highly dependent upon the variable analyzed. Soil test P, K and pH generally had smaller ranges while total N and organic C tended to be larger. Results in this paper indicate that the fundamental field element dimensions are likely to be in the meter or submeter range. At least some of the variables measured in this paper have displayed trends at distances separating measurements of about 10 m. Samples collected at 50 m or larger intervals will miss short distance changes in the values of soil variables. Single samples collected at grid nodes are no better than collecting a single sample from the parent population. In order to describe the variability encountered in the field experiments reported here, soil, plant and indirect measurements should be made at the meter or submeter level.
Prototype equipment has recently been developed (Stone et al., 1996) that can sense and correct nitrogen deficiencies at the submeter level in wheat (Triticum aestivum L.) and bermudagrass (Cynodon dactylon L.). Sensing and variable rate application of other soil nutrients at the submeter level will become practical as real-time sensors are developed for other nutrients. Solie et al. (1996) demonstrated that the range of related measurements of optically sensed plant N fell between 0.70 and 4.46 m, with 1.4 m being common to all transects. They proposed that an area existed in the range of 1.5 x 1.5 m2, that provided the most precise measure of the actual nutritional needs of the crop (fundamental field element). They contended that real-time, variable rate, sensor-applicators should be designed to treat that area. In other work, these authors also demonstrated that large spatial variability existed at the submeter level for both mobile and immobile nutrients in soils (Raun et al. 1998). The authors collected data from 490-0.3 x 0.3 m plots located in 7 x 70 plot arrays, at two locations Burneyville, OK and the Oklahoma State University Efaw farm at Stillwater, OK. Bermudagrass forage yields ranged from 1300 to greater than 10,000 kg ha-1 at the two locations, and soil pH ranged from 4.37 to 6.29 and 5.37 to 6.34. When fertilizer recommendations were based on individual 0.3 x 0.3 m plots, phosphorus recommendations ranged from 0 to 31 kg ha-1 and 0 to 17 kg ha-1, respectively. Potassium recommendations would have ranged from 0 to 107 kg ha-1 and 0 to 108 kg ha-1 at the same sites. These results provide evidence that the inherent variability of soil related variables is in the sub-meter range. If so, sensors and applicators will need to be designed to sense and treat field elements in this range. If current and future sensor and control technologies are to be applied to treating small areas, questions that must be answered include: a) What are the fundamental field element sizes for plant and soil variables of interest; b) What is the magnitude of the error in describing values of variables as the field element treated increases above that fundamental size; and c) Which sampling strategies best account for this variability?
Several researchers have used semivariance analysis to determine the range in which measurements of soil properties were related (Table 1). In general, they sampled at separation distances (distance separating consecutive measurements) much greater than 1 m, and typically, those distances were in the range of 30 to 50 m. Coefficients of variation fell between 16 and 60 %, except pH, which was consistently less than 10 %. Chancellor and Goronea (1994) sampled at a separation distance of 1 m and reported total mineral N coefficients of variation of 34.5 to 66.0 % and the semivariance range of relatedness of 19.5 m. However, their analysis of errors when nitrogen content in a field element was used to predict N levels of field elements at increasing distances from the predictor element suggested that if sampling intervals had been shorter than 1 m, a noticeable amount of application inexactness might have been found.
Previous research indicated that a large number of samples must be collected to describe the true mean of many soil variables. Reuss et al. (1977) calculated that the number of cores required to estimate the mean value of nitrates in irrigated fields were: 82 with an error from the true mean of 15 % and 20 with and error of 26 % for a confidence interval of 90 %. Gupta et al. (1997) reported that the number of samples needed to describe the mean value of three soil variables on 2.2 ha fields at two locations were approximately: 18 samples for NO3-N, P, and K at one location and 33 samples for NO3-N, 11 samples for P, and 11 samples for K at the second location using a grid sampling strategy to collect samples. Paz et al. (1996) found in a 1.8 ha field that: pH could be estimated with 10 % error from the true mean at 95 % confidence interval with two samples and soil organic C estimated with 16 samples. Mean extractable nutrients required 16 to 20 samples to estimate the relative mean with 20 % error.
Semivariance analysis has been discussed in most journal articles to define
relatedness between samples of spatially varying soil or plant variables.
However, because in this paper the procedure is being applied to contiguous
small (0.3 x 0.3 m plots) rather than measurements at separation distances of
30 or more meters, the procedure will be reviewed in detail. Semivariance
analysis is a geostatistical analysis method used to estimate the distance or
range over which samples of a regionalized variable have related variances (Royle
et al., 1980). Regionalized variables are variables whose positions in space
and time are known (Woodcock, et al., 1988). These variables have similar
values (low variances) at close, (hence regional) locations, but become
increasingly disparate as the distance separating the measurements increases
(variance increases). The semivariance,
of all samples separated by a distance h is calculated by equation 1:
with semivariances calculated for all separation distances where is the sampling increment and L is the longest separation distance at which 25 to 30 pairs of points can be obtained (Journel and Huijbregts, 1978). Semivariances are customarily plotted as a function of h and are termed semivariograms. It should be noted that the terms separation distance and lag are used interchangeably in spatial statistics literature.
There are four semivariance characteristics which are important for determining maximum field element length: the nugget, sill, range, and trend or gradient (Royle et al., 1980). The nugget is the semivariance at zero separation distance. Nugget semivariance is random and is a function of the variability within the sample. The sill is the upper limit of the semivariance, and the range defines the minimum separation distance of statistically independent pairs of samples. Occasionally, the ratio of the nugget to the sill (N:S) is calculated, and this ratio is normally expressed as a percentage (Isaaks and Srivastava, 1989). This ratio has been termed the relative nugget effect. Values of the relative nugget effect near 1.0 indicate that a large degree of the variability is associated with the within sample measurements, and that relatedness between spatially separated measurements is limited. A relative nugget effect near zero indicates that the relatedness of spatially separated measurements within the range is strong. The range represents the zone of influence of the sample (Knighton and Wagnet, 1987). Beyond the range the semivariance becomes approximately equal to the population variance of the measured variable. An underlying assumption of the analysis is stationarity. Assumptions of stationarity include: the expected value of all samples exists and does not depend on the separation distance, h, and the sill semivariance is independent of the separation distance. If, at some separation distance beyond the range, the semivariogram suddenly turns upward in a parabolic type curve, there is a strong underlying trend or gradient in the measured variable (Royle, et al., 1980). This separation distance, which the authors have termed the departure distance (Solie et al., 1996), is not normally calculated in semivariogram analysis. However, if the departure distance is greater than the range then a sill exists. Sills can be nested within a semivariogram (Hohn, 1988), and the sill with the minimum range distance can be used to define the size of the fundamental field element. Samples beyond the range are statistically independent. If sensing and variable rate application are to be performed on only areas with related measurements, then the maximum field element size for the measured variable should be less than the range.
Many semivariograms appear to oscillate about the sill. The oscillation, termed pseudocycling, is defined as the apparent periodic cycling or oscillation of the magnitude of the variable over distance, and is a common phenomenon with minerals (Hohn, 1988). Normally, changes in magnitude are random or aperiodic even though they appear periodic. However, in arable soils, it is possible for them to be periodic if induced by tillage, fertilizer application, or by other field operations.
Measurements with spacing equal to the range are unrelated. Han et al. (1994), proposed that some distance less than the range would be more appropriate to define the region containing closely related samples. They proposed the mean correlation distance (MCD) as the appropriate distance. The mean correlation distance was derived from the normalized complement function:
where co is the nugget semivariance and c equals the sill semivariance minus the nugget (co). The nugget to sill ratio can be defined using Han et al.s (1994) notation as:
MCD = (4)
where r is the range.
Objectives: Analyses were performed on plant and soil data presented by Raun et al. 1998) to:
1. Determine the range within which measurements were related.
2. Evaluate the error associated with various size field elements.
3. Evaluate the effectiveness of alternative soil sampling schemes in estimating the true means of the data sets.
Materials and Methods
Five soil variables, total soil N, extractable phosphorus and potassium, organic carbon, and pH and two plant variables, forage nitrogen and biomass, were measured in 490-0.3 x 0.3 m plots at two locations. Experiments were conducted on two established bermudagrass pastures located at the Efaw Experiment Station farm, Stillwater, OK and at Burneyville, OK. The soil at Efaw was a Norge loam; fine-silty, mixed thermic Udic Paleustoll, and at Burneyville, a Minco fine sandy loam; coarse-silty, mixed thermic Udic Haplustoll. The site at Efaw was mowed each year, but was not pastured or harvested for hay. At Burneyville, the site was part of a large pasture that was annually grazed. No fertilizer had been applied at either location in at least five years. At each location, a visually homogeneous 2.13 x 21.3 m area was selected for intensive soil sampling. Each area was subdivided into 490-0.3 x 0.3 m plots. Eight soil cores (13 mm in diameter and 0 to 150 mm deep) were collected and composited from each 0.3 x 0.3 m plot. Raun et al. (1997) presented a complete description of the soil test procedures, and analyses of sampling and laboratory error. Data were subjected to a standard descriptive statistical analysis.
Semivariance Analysis: Semivariance analysis was used to estimate the range over which samples of the five soil variables and two plant variables were related. The semivariance, , of all samples separated by a distance h was calculated by equation 1. Semivariances were calculated only in the direction of the longest (21.3 m) axis of the experiments. The maximum separation distance was 60 cells (18.29 m). At that distance, semivariances were calculated from 70 difference measurements. Data were examined and outliers deleted that affected the semivariograms, using the procedure of Isaaks and Srivastava (1989).
Semivariance statistics including: the nugget, sill, N:S ratio, range of relatedness were calculated using the following procedure. Semivariances were calculated (equation 1) and plotted as scatter diagrams. The plots were visually examined to locate sills. Data files were clipped at the point where the semivariance data departed from the sill, because data beyond that separation distance shifted the sill semivariance. A linear-plateau (linear to a sill) function was fitted to the data (Solie et al., 1996), when the relationship between the semivariance and separation distance was linear. The range was defined as the distance where the curve changed from a positive slope linear curve to the sill. The best fit nonlinear peak function in the curve library of the program 'TableCurve' (Anonymous, 1997) was fitted to the data when the curve was nonlinear. The peak curve did not always reach a plateau within the apparent range, increasing slightly (less than 0.1 %) with each additional separation distance increment. In these cases, the range was defined as the distance at which the change in semivariance was less than 0.01% of the sill value with an incremental change in h. This approach for fitting nonlinear curves was different from normal procedures for fitting standard curves to the data (Hohn, 1988; Knighton and Wagnet, 1987), but the primary reason for standard curves is to facilitate kriging. In this case, curve fitting was used only to determine the range, so non-standard curves, which often fit the data better were used. This procedure closely paralleled that used in Solie et al. (1996).
Field Element Size Effect on Error: An alternative to semivariance analysis for determining the relationship between field element size and the measured value of the soil or plant variable in each of the constituent plots is to compare the error between the average value of that variable for a given sized field element with the measured value of each 0.3 x 0.3 m plot contained within that element. To calculate that error, each data set was subdivided into the field elements shown in Table 2 and Figure 1 a. Field element size was chosen so field elements of increasing size evenly divisible into the 6 x 66 plot array element. The mean error was defined as the average value of the difference between the average value of the field element and the actual value of that variable for each of the plots constituting the field element.
Sampling Strategies for Determining Mean Values: Three sampling strategies were evaluated to find the best sampling strategy for determining the true mean of data with submeter variability: random sampling, fixed-interval grid sampling, and stratified-random sampling. The random sampling strategy was implemented by randomly selecting 35 replications of 1, 2, 5, 7, 10, and 14 samples (plots) from the 7 x 70 plot experimental areas (Table 3 and Figure 1 b). An Excel macro was written implementing the Monte Carlo sampling procedure to randomly select, with equal probability, and without replacement the plots needed to build 35 replicate sampling sets. By sampling without replacement, no plot was sampled more than once in any replication.
To implement the fixed-interval grid sampling strategy, plots for sampling were located at predetermined fixed intervals (Figure 1 c). The first grid sample was randomly located within a variable area ranging from 7 x 70 cells for 1 sample to 7 x 5 cells for 14 samples located at one end of the experimental area (Table 3). Sampling intervals ranged from 70 cells for one sample collected from the experimental area to 5 cells for 14 samples. The location of the first grid sample was randomly determined using the Monte Carlo procedure. All additional sample plots for that replication were located along the same transect at the appropriate interval. Sampling was performed without replacement and with 35 replications.
With stratified sampling, zones were located at fixed intervals, but the sample plot location was randomly selected within each zone (Figure 1 d). Zone size was 7 x 3 plots, with the 7 cell dimension parallel to the narrow side of the experimental area. Plots were randomly sampled in each zone. Zone spacing was a function of the number of samples and was identical to the interval spacing used in the grid sampling strategy. Sampling was done with replacement. Plots were sampled more than once when only one sample was collected per replication.
One approach to estimating the number of required samples is to conduct a replicated computer experiment in which plots (samples) are randomly selected until all plots contained in the experiment are measured, at which point the error from the true mean will be zero. The computer sampled experiment can be replicated to develop a distribution of error in measurement from the true mean of the experiment for each sample size. Confidence intervals can be calculated for each sample size distribution using standard procedures (Snedecor and Cochran, 1980). These data can be used to define the error in estimating the true mean of a field element of any size and to fix the confidence interval about that estimate. To implement this procedure, data at both locations were randomly sampled until all plots were sampled. Sampling was replicated 35 times. The sample sizes required to reach 10 %, 5 %, or 2 % error from the experimental area (2.31 x 23.17 m) true mean for the 5 and 95 % confidence intervals were calculated for 35 of the replications.
Results and Discussion
Descriptive Statistics: The coefficients of variation for soil and plant analyses performed in this research fell midway within the data reported by several researchers (Table 4 and 1, respectively). Only the pH exhibited low variability with a coefficient of variation (CV) of 3.2% at Efaw and 4.2% at Burneyville, where CV is used to describe the amount of variation in a population (Snedecor and Cochran, 1980). All other variables had high variability (CV > 30 %) at one or both locations.
Semivariance Analysis: Semivariograms of the data displayed unique responses (Fig. 1 and 2). Six of the fourteen semivariograms (total soil N, soil organic carbon, and pH at Burneyville and soil organic carbon, forage N , and biomass at Efaw; Fig. 2 a, g, h, and i and Fig. 3 b and d) were nonlinear between the nugget and sill. Semivariograms for phosphorus at both locations, soil organic carbon at Efaw, and biomass at Burneyville displayed two or more clearly defined sills. Drift was apparent in these semivariograms as well as total soil N semivariograms at both locations. There was at least some pseudocycling in most semivariograms. Sill length varied, but in many of the semivariograms, semivariances appeared to depart from the sill at about 10 m. At that distance the underlying drift or trend in the soil or plant variable began to dominate the semivariance.
The semivariance curve for pH at Efaw had a region between the nugget and peak semivariance that was divided into two distinct zones described by different curves (Fig. 2 j). There was no sill linking the zones, but there was a sigmoidal type transition. We considered this semivariance curve to have two separate and distinct ranges. This is contrasted with the Efaw forage N semivariogram (Fig 3 b) where a single monotonic curve was fitted to the data even with apparent inflections in that data.
The relative nugget effect, N:S ratio, ranged from a high of 0.729 for total soil N at Efaw to a low of 0.042 for pH at Burneyville (Table 5). The high value for total soil N at Efaw indicated that most of the variability was associated with the nugget and that the degree of relatedness among samples was weak. The low value of the Burneyville pH semivariogram nugget value was 0001302 and N:S = 0.042 indicating a high degree of relatedness among samples within the semivariance range. The semivariance ranges of the seven variables ranged from 1.04 to 6.10 m. When nugget variability was factored into the estimate of the range of relatedness among samples by calculating the MCD, the distance was much smaller ranging from 0.35 m for phosphorus at Burneyville to 1.95 m for forage N at Burneyville. The existence of multiple sills implied that additional ranges and MCDs, encompassing greater variability existed. Four semivariograms exhibited two or more clearly defined sills: Burneyville phosphorus - range = 16.18 m and MCD = 6.98 m; Burneyville biomass - range = 4.67 m and MCD = 1.34 m; Efaw phosphorus range = 12.23 m and MCD = 4.20 m; and Efaw pH - range = 6.54 m and MCD = 2.12 m.
Field Element Size Effect on Error: One way of examining the effect of field element size on error from the true value of the measurement is to assume a sensor exists that can accurately and precisely measure the value of the measured variable averaged over a small (<5 x 5 m) field element. Although these sensors are only now being developed, there is a high probability that they will be available in the future. Assuming that a sensor exists, the following question can be posed: What is the error from the measured value of an individual plot (or cell) of the sensor measured value for the field element? As the field element size decreases the error between the values should approach zero. Plot size in these experiments was 0.3 x 0.3 m. Calculations were performed to determine the mean errors between the measured values of each plot within a field element and the average value of that field element (Table 6). Mean error decreased about 50 % as the field element area measured decreased from the 7 x 70 plot array to 2 x 2 plot arrays. The greatest decrease occurred with variables which had an underlying drift in the mean value or had large fluctuations in values because of pseudocycling. This was the case for phosphorus at Efaw, where the mean value of the error decreased from 21.9 % for a field element the size of the experimental area (490-0.3 x 0.3 m plots) to 8.2 % when the field element encompassed 4 plots. The decrease in error was least where the variable exhibited no drift over a relatively large distance, e.g. total soil N at Efaw (Fig. 2 a), where the mean value of the error decreased from 10.3 % for a field element equal to the experimental area to 6.3 % for field elements equal to 4 plots. Mean error could be very large, 46.2 % for phosphorus when sensed over a 6 x 66 field element, to negligible, 1.3 %, when sensed over the 2 x 2-4 plot field element for pH at Efaw. The greatest benefit occurred when reducing the field element size for phosphorus and potassium. Reducing the field element size for total soil N did not produce equivalent benefits, because the previous crop history suggested that total soil N would be uniformly distributed at comparatively low levels.
An underlying assumption of any semivariance analysis is the existence of a fundamental field element. Both the range and the MCD have been used to define that area. Solie et al., 1996 concluded from semivariogram analysis of optical sensor data that the fundamental field element should be about 1.5 x 1.5 m. Another possible definition is the rooting area of the individual plants. The rooting distance of bermudagrass has not been reported, but the authors have observed pronounced striping when solution N fertilizer was applied with streaming nozzles spaced 0.25 m apart. For wheat, the rooting distance is approximately 0.3 m. If the rooting distance definition is used to define field element size, sensing 0.61 x 0.61 m area (2 x 2 plot array) produced mean errors from 1.3 to 30 % or greater depending on the sensed variable (Table 6). If the MCD is accepted as the definition, the fundamental field element size will be somewhere between the rooting distance of wheat to twice that distance depending on the plant or soil variable at both locations.
Sampling Strategies for Determining Mean Values: Number of samples and sampling strategy were highly significant factors affecting error from the true mean of the 7 x 70 plot field element for all variables at all locations (Table 7 a, b). The interactions were significant (0.05 level) at all locations and for all variables except total soil N and pH at Burneyville and soil organic carbon, forage N , and biomass at Efaw. Both random sampling and the stratified-random sampling strategies produced significantly lower error than fixed-interval grid sampling for all variables at both locations with the following exceptions: random sampling for forage N at Efaw and total soil N at Burneyville and stratified-random sampling of biomass at Burneyville. Some interactions occurred between spacing and sampling strategy. In a few instances, either random sampling or stratified-random sampling was better than the other. However, there was no clear trend favoring either random sampling or stratified random sampling. When interactions were examined, both the random and stratified-random sampling strategies were better at estimating the mean value than fixed grid sampling, in nearly all cases when two or five samples were collected.
Results of the analysis where the data were randomly sampled until all plots in the experiments were sampled defined the range of samples sizes needed to reach the target measurement error. Of the measured variables, pH was the most uniformly distributed at both locations with the lowest CV. Four samples were required at Burneyville and three at Efaw to estimate the average value of pH with 2 % error for the experimental area with 95 % confidence (Table 8). Total soil N, potassium, and forage N could be estimated with 10 % error at the 95 % confidence interval with 4 to 11 samples. At Efaw, soil organic C and phosphorus could be estimated with 10 % error and 95 % confidence by collecting 2 and 8 samples respectively. But the number of samples required increased to 39 samples for phosphorus and 17 for soil organic C at Burneyville. Because of the high variability, biomass required 26 and 31 samples to reach 10 % error at the two locations. Of the variables other than pH, only forage N at both locations, soil organic C at Efaw, and total soil N at Efaw could be estimated with 5 % error at the 95 % confidence interval with 16 or fewer samples. These data indicate that even in a 2.13 x 21.3-m area, 10 % error is the best that can be expected for 16 samples while soil phosphorus may require more samples. These results agree with data reported for much larger areas (Gupta et al., 1997, Paz et al., 1996, and Reuss et al. 1977).
Discussion: Results of the two experiments reported in this paper when compared with results reported previously for much larger areas and non-contiguous sampling zones raise serious questions about sensing and sampling strategies. Coefficients of variation and estimates of number of soil samples required to reach acceptable error for reported data were comparable to C.Vs and sample sizes calculated for the two experiments reported in this paper. Although it is impossible to make direct comparisons between the two experiments reported in this paper and the results from other papers, an inference arises that variability in a 20 x 20 m2 area can be the same as that for one hectare and larger areas. Although other papers document the existence of sills at ranges greater than 10 m, results in this paper indicate that the fundamental field element dimensions are likely to be in the meter or submeter range. At least some of the variables measured in this paper have displayed trends with increasing semivariances of measurement collected greater than 10 m apart. Samples collected at 50 m or larger intervals will miss shorter distance changes in the values of the soil variables. If C.V.s reported in the literature are comparable to the C.V.s found in this experiment, then single samples collected at grid nodes are no better than collecting any single sample from the parent population. Multiple samples collected in the vicinity of a grid node will better estimate the mean value of the variables at the node, but will convey no information about changes in those values between nodes. Interpolation schemes, no matter how sophisticated, that use data collected at 50 m intervals are no better at estimating the true value of a 1 m2 fundamental field element than randomly sampling the area and using the mean value of the measured variable. When sampling at these intervals, it is only by chance that the error in fertilizer application can be reduced from that based on random sampling the entire area. Oppenheim and Willsky (1983) should be consulted for digital signal processing, discussion of sampling frequency, aliasing, hidden oscillations and their relationships to control systems. If fertilizers are to be variably applied based on measured available nutrients, sensors must be designed that measure target variables at the meter or submeter level. Until these sensors are developed, fertilizer applications can only be made based on the average value of the nutrients in the management area. The accuracy of this application will be only as good as the precision at which the true mean value is estimated. The degree of the meter level variability in agricultural fields must be defined if the objectives of precision application of plant nutrients are to be realized.
The authors thank the Oklahoma
Agricultural Experiment Station, the Soil Fertility Research and Education
Advisory Board and The Samuel Roberts Noble Foundation, Inc. for supporting
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Figure 1. Experiment layout showing relative field element sizes used in calculating error from true plot (cell) values of the average value of the element (Fig 1a); random sampling scheme for 10 samples (Fig. 1b); fixed interval grid sampling scheme for ten samples (Fig. 1c); and stratified-random sampling scheme for ten samples with each sample randomly select from a 3 x 7 plot array and each array spaced at a fixed interval (Fig 1d).
Figure 2. Semivariograms for five soil variables at two locations.
3. Semivariograms for two plant variables at two locations.